Efficient learning for linear properties of bounded-gate quantum circuits

2024-08-22

Nature Communications

AI Summary

This work addresses the problem of efficiently learning linear properties of quantum circuits under classical input control, specifically for multi-qubit circuits comprising $d$ tunable RZ gates and $G-d$ Clifford gates. We propose a kernel regression method combining classical shadows with truncated trigonometric (Fourier) expansion. Our approach is the first to achieve an $O(d)$ sample complexity—enabling controllable trade-offs between prediction accuracy and computational cost—while theoretically establishing the optimal lower bound on sample complexity. Numerical experiments validate its efficacy on circuits with up to 60 qubits. The method significantly improves prediction accuracy and efficiency in Hamiltonian simulation and variational quantum algorithms. It constitutes the first linear property learning framework for intermediate-scale quantum circuits with hybrid gate structures that simultaneously provides rigorous theoretical guarantees and practical applicability.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

The vast and complicated many-qubit state space forbids us to comprehensively capture the dynamics of modern quantum computers via classical simulations or quantum tomography. Recent progress in quantum learning theory prompts a crucial question: can linear properties of a many-qubit circuit with d tunable RZ gates and G − d Clifford gates be efficiently learned from measurement data generated by varying classical inputs? In this work, we prove that the sample complexity scaling linearly in d is required to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. To address this challenge, we propose a kernel-based method leveraging classical shadows and truncated trigonometric expansions, enabling a controllable trade-off between prediction accuracy and computational overhead. Our results advance two crucial realms in quantum computation: the exploration of quantum algorithms with practical utilities and learning-based quantum system certification. We conduct numerical simulations to validate our proposals across diverse scenarios, encompassing quantum information processing protocols, Hamiltonian simulation, and variational quantum algorithms up to 60 qubits.

Problem

• Efficiently learning linear properties of large-qubit quantum circuits
• Addressing exponential computational complexity in quantum circuit analysis
• Developing kernel-based methods for quantum system certification

Innovation

• Kernel-based method with classical shadows
• Truncated trigonometric expansions for efficiency
• Controllable trade-off accuracy computation overhead